Correlation coefficient: the last block of statistical foundation
Correlation has already been mentioned in
Statistical measures and their geometric roots
Properties of standard deviation
The pearls of AP Statistics 35
The pearls of AP Statistics 33
The hierarchy of definitions
Suppose random variables are not constant. Then their standard deviations are not zero and we can define their correlation as in Chart 1.

Chart 1. Correlation definition
Properties of correlation
Property 1. Range of the correlation coefficient: for any
This follows from the Cauchy-Schwarz inequality, as explained here.
Recall from this post that correlation is cosine of the angle between
Property 2. Interpretation of extreme cases. (Part 1) If
(Part 2) If
Proof. (Part 1)
(1)
which, in turn, implies that
Plugging this in Eq. (1) we get
The proof of Part 2 is left as an exercise.
Property 3. Suppose we want to measure correlation between weight
Proof. One measurement is proportional to another,
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